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The Principle of General Covariance This principle was used by Einstein as a guide in the formulation of General Relativity The form of the physical laws must be invariant under arbitrary coordinate transformations

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One question: is there another kind of invariance of the equations of General Relativity? One kind of invariance that has attracted the attention of theoreticians in other branches of physics is the so-called conformal invariance This concept first arose with H. Weyl, in 1919, in his attempt to unify gravitation and electromagnetism

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One of the simplest examples is Weyl conformal gravity: Interest in this new form of invariance has led to the investigation of conformal gravity theories Conformal transformation

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All these gravitational theories are fundamentally different from general relativity and give predictions that are not consistent with the observational facts Weyl conformal gravity leads to fourth order derivative in the field equations

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An interesting fact is that… change the geometric description of space-time Riemann Weyl if we We have a new fundamental group of transformations

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These are called Weyl transformations They include the conformal group as a subgroup What is Weyl geometry ?

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In Weyl geometry, the manifold is endowed with a global 1-form Riemannian geometry

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Weyl integrable geometry We have a global scalar field defined on the embedding manifold, such that A particular case is

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The interesting fact here is that... Consider the transformations We can relate the Weyl affine connection with the Riemannian metric connection

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...geodesics are invariant under Weyl transformations ! The concept of frames in Weyl geometry The Riemann frame General Relativity is formulated in a Riemann frame, i.e. in which there is no Weyl field

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Riemann frame First question: Can we formulate General Relativity in an arbitrary frame? Second question: Is it possible to rewrite GR in a formalism invariant under arbitrary Weyl transformations?

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The answer is... Yes! The new formalism is built through the following steps: First step: assume that the space-time manifold which represents the arena of physical phenomena may be described by a Weyl integrable geometry

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We need two basic geometric fields: a metric and a scalar field Second step: Construct an action S that be invariant under changes of frames Third step: S must be chosen such that there exists a unique frame in which it reduces to the Einstein- Hilbert action

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Fourth step: Extend Einstein’s geodesic postulate to arbitrary frames. In the Riemann frame it should reproduce particle motion predicted by GR Fifth step: Define proper time in an arbitrary frame. This definition should be invariant under Weyl transformations and coincide with GR’s proper time in the Riemann frame

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In n-dimensions the action has the form What happens if we express S in Riemannian terms ?

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For n=4

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In the vacuum case and vanishing cosmological constant this reduces to Brans-Dicke for w=-3/2 However the analogy is not perfect because test particles move along Riemannian geodesics only in the Riemann frame

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Proper time: we need a definition invariant under Weyl transformations In an arbitrary frame it should depend not only on the metric, but also on the Weyl field The extension is straightforward:

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Under change of frames null curves are mapped into null curves Consequences: The light cone structure is preserved Causality is preserved under Weyl transformations

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This change of perspective leads, in some cases, to new insights in the description of gravitational phenomena Gravity in the Weyl frame

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Variation Principles In an arbitrary Weyl frame variations shoud be done independently with respect to the metric and the scalar field In four dimensions this leads to This is General Relativity in disguise!

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In this scenario the gravitational field is not associated only with the metric tensor, but with the combination of both the metric and the geometrical scalar field We can get some insight on the amount of physical information carried by the scalar field by investigating its behaviour conformal solutions of general relativity

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In the Riemann frame the manifold M is endowed with a metric that leads to Riemannian curvature, while in the Weyl frame space-time is flat. Consider, for instance, homogeneous and isotropic cosmological models These have a conformally flat geometry There is a frame in which the Geometry becomes flat (Minkowski)

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This leads to quite a different picture. For instance The Weyl field will be given by

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Another simple example is given by some Brans-Dicke solution. For instance, consider O`Hanlon-Tupper cosmological model and set w=-3/2 It is equivalent to Minkowski spacetime in the Riemann frame.

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Conclusions: There is no unique geometrical formulation of General Relativity As far as physical observations are concerned all frames are completely equivalent Is this kind of invariance just a mathematical curiosity or should We look for a “hidden symmetry”?